Even as the Hudson was flooding the four-mile long, 550-acre park, Mr. Heatherwick presented an idea for a new pier that would jut into the river at West 13th Street and replace Pier 54, which was closed the year before, after it started to collapse.

The shuttered 875-foot pier was where survivors of the Titanic were brought in 1912. More recently, it was a popular place for outdoor concerts, screenings and exhibitions.

“It was very cool looking,” recalled Diana L. Taylor, who is chairwoman of the Hudson River Park Trust’s board of directors, referring to Mr. Heatherwick’s design.

“It showed the amount of work he put into it — his imagination,” said Madelyn Wils, the president of the trust, a public benefit corporation that oversees the development and operation of the park. “That’s what struck everyone.”

The donation, the largest to a public park in New York City’s history, will pay for a differently designed new pier: a rotated square detached from the bulkhead, with landscaped knolls, a roomy amphitheater and other performance areas.

“I have always liked the idea of public art and public places, and that’s just been growing in me over decades,” Mr. Diller, 73, said in a phone interview on Tuesday. “It isn’t going to cure world disease. But it’s totally worthwhile.”

The plan was announced with much fanfare in November. Pier 55, as the new pier-cum-culture-island is called, was showered with supportive sound bites from the governor’s office and City Hall.

The talks surrounding the plan had been so secretive, though, that the announcement was a surprise to even the closest observers of the park. And not everyone embraced the way the new 2.4-acre pier — dubbed “Diller Island” by some of its critics — was being presented: basically, as a fait accompli.

Assemblywoman Deborah J. Glick, a Democrat whose district includes part of the park, was among those who found the new plan problematic.

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The shuttered Pier 54.CreditHiroko Masuike/The New York Times

In 2013, she had signed off on an amendment to the original state legislation that created the park permitting Pier 54 to be “reconstructed outside of its historic footprint.” But Ms. Glick said she agreed to the change after hearing from Ms. Wils about the need to improve egress for concertgoers. She said there was no mention then that the trust was deep in talks with Mr. Diller to create the cultural site.

“They cast it as a redevelopment of the pier from narrow and long to short and wide,” Ms. Glick said. “There was never any clarity that they were involved in negotiating a major new pier. It’s an entirely new development and a major intrusion in a sensitive marine environment.”

Ms. Glick said she resented having been kept in the dark. “Obviously, I was concerned that we were never provided the full explanation of what they wanted,” she said. “They were less than candid.”

Big gifts to parks have a habit of turning into lightning rods. When John A. Paulson, the hedge fund manager, pledged $100 million to Central Park in October 2012, advocates for smaller, scruffier parks around the city cried foul, saying the gift highlighted inequities in the park system.

And when Joshua P. Rechnitz, a wealthy cycling aficionado, said he would give $40 million to build a field house and velodrome in Brooklyn Bridge Park, it was viewed as a white elephant, and he eventually withdrew the plan.

Similarly, the $113 million for Pier 55, for which Hudson River Park still has only an oral pledge from Mr. Diller, has put the spotlight on the park’s longstanding financial problems, embodied by the deterioration of Pier 40, at the foot of Houston Street. A campaign a few years ago to allow luxury housing at the pier as a way to generate income was quashed by Ms. Glick. She grudgingly signed off on an air rights amendment that was part of the 2013 package of legislation designed to help the park financially. That angered preservationists who worry about the potential for bulkier, taller development.

Those same preservationists look at the money for the Pier 54 replacement with a critical eye. Not only is the private sector contributing more than $100 million, but the city is tossing in another $17 million. The money, combined, would just about pay for repairs at Pier 40.

“But it does seem a little curious that there’s this vast pool of largely private — but also public — money for this addition to the park,” he said. “I know Pier 54 was always supposed to be replaced, but this is vastly grander than what was planned. How can we claim that there is not private or public money for completing the parts of the park that need to be done?”

Hudson River Park was created in 1998 under Gov. George E. Pataki and Mayor Rudolph W. Giuliani, both Republicans, in what was hailed as an innovative city-state partnership. The idea was to transform the largely derelict Hudson River waterfront from Battery Place to West 59th Street into a continuous greensward, with a bicycle path and piers for recreation. The city and the state would finance its construction, but the park would be maintained with revenue from commercial development within its borders.

Initially, officials said it would cost about $300 million to build the park, which was expected to be completed in 2003. Within a year, that had jumped to $370 million. The city and the state have already spent more than $330 million in capital funds on the park, with the federal government allotting another $110 million, and it is still only 70 percent complete. Since the financial crisis in 2008, the city and state have offered little new capital money.

As Hudson River Park struggled, Central Park was turned into a burnished gem, the High Line was transformed from an abandoned rail track into one of the city’s most popular tourist attractions and Brooklyn Bridge Park leapfrogged ahead as the city’s waterfront greenway. But that park, on the East River, is about one-seventh the size of Hudson River Park. Its piers were also in better shape. And Brooklyn Bridge Park benefited from Mayor Michael R. Bloomberg’s laser focus and largess, with consistent annual infusions of capital money.

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A rendering of Pier 55, which is expected to open in about four years.CreditPier55, Inc./Heatherwick Studio

Even unfinished, Hudson River Park is an emphatic success. Last year, millions of visitors went there to fish, kayak, skateboard, salsa dance and learn about the estuary. Bicyclists and joggers whiz past rose mallows, northern bayberry and milkweed, the Hudson River a luminous backdrop. But the park has also struggled with upkeep, as the expense of caring for structures built on or over water has turned out to be far greater than anticipated, with piers and bulkheads subjected to tides, winds, fungi and marine borers.

Many of the park’s troubles have centered on Pier 40. The 15-acre pier, where the trust has its offices, was supposed to be a revenue producer for the park — one of several to host commercial development.

But a number of development proposals — big-box retail, an entertainment complex, a circus — collapsed under the weight of community opposition. As requests for proposals came and went, Pier 40’s 3,000 concrete pilings were deteriorating, along with its roof. (The latest estimate to repair the pilings: $104 million.) The ailing structure scared off potential developers, reluctant to take on the repairs.

The deterioration also threatened the pier’s current uses — as a parking garage that generates $5.5 million a year (the single largest source of revenue) and a pair of popular athletic fields. In the past few years, the trust had to move hundreds of cars to avoid falling debris. It also closed a play area in the southwest corner of the doughnut-shaped pier, where the pilings were deemed unsafe.

Then Hurricane Sandy devastated the park, its floodwaters frying miles of electrical cable and causing $32 million in damage. Weeks without power turned into months. It wasn’t until 20 months after the storm that the park’s lights all blinked back to life.

In 2013, park officials won permission from state lawmakers to sell air rights in order to finance work at Pier 40 and other projects. A proposal to sell those rights to the owner of the sprawling St. John’s Terminal Building, across the West Side Highway from the pier, is now wending through the city’s planning process.

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Madelyn Wils of the park’s trust.CreditHiroko Masuike/The New York Times

Given Pier 40’s well-publicized woes, the shutdown of Pier 54 in 2011 went largely unnoticed beyond the immediate neighborhood. But park officials, sensitive to the pier’s role as a performance site, immediately set out to find a solution.

Mr. Diller’s IAC headquarters, with its sinuous white facade designed by Frank Gehry, looms over the West Side Highway at 18th Street, a few blocks north of Pier 54. Mr. Diller, a San Francisco native who has long lived in New York, bicycles in the park, as do his employees. He and Ms. von Furstenberg have donated large sums to Carnegie Hall, Signature Theater, the Whitney Museum and the future Culture Shed at Hudson Yards, as well as to Central Park and the High Line.

Three separate gifts to the High Line, totaling $35 million, put the couple in the first ranks of park philanthropists. “Each gift had an extraordinary impact,” said Joshua David, a founder of Friends of the High Line. “Diane was in this neighborhood before the High Line project took off. They’ve hosted a number of events that have helped us bring in many new donors.”

Indeed, officials of Hudson River Park were inspired by the High Line’s success in revving up its own fund-raising efforts. “We thought, ‘Well, someone who is sort of obvious to go talk to is Barry Diller,’ ” said Ms. Taylor, who is the companion of former Mayor Michael R. Bloomberg. “He and his wife have been extremely generous for the arts, but especially the High Line. So we thought, ‘O.K., you don’t ask, you don’t get.’ ”

In early 2012, Ms. Taylor and Ms. Wils met with Mr. Diller in his office, showing him an early sketch for an amoeba-shaped pier that Ms. Wils had conceived. “Barry was cool at the beginning,” Ms. Wils recalled. “It was not like, ‘Oh, I love this. I want to do this.’ ”

He explained that with Ms. von Furstenberg’s children settled in Los Angeles, the couple were thinking of moving their philanthropy there, Ms. Wils recalled. But he listened to their request — $35 million for a new pier, which would supplement the anticipated government contribution. And he agreed to follow-up meetings. Eventually, his vision expanded and he brought in other arts world figures to brainstorm ideas. In addition to Mr. Rudin, these included George C. Wolfe, the former producer of the Public Theater; Stephen Daldry, the director and producer; and Kate Horton, a British theater executive.

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Diana L. Taylor, also of the park’s trust.CreditHiroko Masuike/The New York Times

The design also became a paramount focus. “He said, ‘Look, if I’m going to be interested in this, we have to do something ambitious architecturally,’ ” Ms. Wils remembered.

In 2013, Mr. Diller summoned Ms. Wils and Mr. Heatherwick to a field in Islip, on Long Island, to see a mock-up of the pier, using ropes to delineate the amphitheater, a piazza and the width of paths. The design now resembled the one that was presented to the public last fall: an offshore pier turned diagonally to the bulkhead, its elevation ranging from 15 to 70 feet above the water. Raising the pier above the new, post-Hurricane-Sandy flood plain allowed light underneath to improve fish habitats and offered views of the river and the skyline.

On Long Island, Mr. Diller was hoisted up over the field in a cherry picker. “It was something of a revelation,” he said. “It was the most impressive thing to see from various angles how you would look out to the city and imagining how the city would look back to you.”

Pier 55, unlike the 16 other piers that have been rebuilt in the park, would not use the same footprint as the one it will replace, Pier 54. Rather, the new pier would be built slightly to the north, with a different geometry, which would require 300 new concrete columns to penetrate the undisturbed riverbed in a part of the Hudson designated as an estuarine sanctuary.

Tobi Bergman, chairman of Community Board 2, which passed a resolution in January in support of Pier 55, said he was not bothered by the secrecy of the deal. Philanthropists tend not to share their thought process with the public about major gifts, he noted. “These kinds of things just do not get negotiated in public,” he said. “It’s not possible.”

The community board sought assurances that ticketed events would be affordable and that the public would not be unnecessarily excluded from the pier during performances. The final lease agreement, approved in February by the trust, sought to assuage those concerns, as well as others about the long-term finances. The pier is expected to be finished in late 2018 or early 2019, and Mr. Diller has agreed to run it and cover operating expenses for 20 years. The lease agreement also lowered the pier’s highest point to 62 feet, to make it handicap accessible.

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Assemblywoman Deborah J. Glick, a Manhattan Democrat whose district includes part of the Hudson River Park.CreditMichelle V. Agins/The New York Times

Recently, Ms. Wils called the final plan a “home run.”

In addition to Pier 40, the trust still needs to finish Pier 57, also designated for commercial development, at West 15th Street. Park officials are working with the developer Youngwoo & Associates and RXR Realty on a retail and office complex there. Piers 26 and 97 also need to be completed for public use, along with a five-acre parcel on landfill at Gansevoort Street that the Sanitation Department has been ordered to relinquish.

Then there are 15 blocks of esplanade and upland areas, between West 29th and 44th Streets, that need new landscaping and other capital work. Ms. Wils estimates the cost of finishing the public portions of the park — beyond Pier 54 — at $175 million.

Officials at the trust say they hope to strengthen the relationship with the city and the state to secure additional capital money. “We need to finish the park,” Ms. Wils said, making the case that Hudson River Park had attracted billions in real estate investment to the west side of Manhattan.

As to the idea that Mr. Diller’s money would be better used to repair Pier 40, Ms. Wils said, “If anyone knows a donor willing to cover the cost of pile repairs, we’re all ears.”

For fans of the park, the gift from the Diller-von Furstenberg foundation represented more than the revival of Pier 54. It was a statement that Hudson River Park deserved to be in the same league as the city’s other signature, showy spaces.

“It’s going to put the park on the map in ways that it has not been,” said Mr. Bergman, the Community Board 2 chairman, who has also organized the parents of athletes to speak out for Pier 40. “It hasn’t been a glamorous park. But this is New York City, and what’s important in New York is glamour.”